cat017-appendix-a

EUROPEAN ORGANISATION FOR THE SAFETY OF
AIR NAVIGATION
EUROCONTROL
ASTERIX Part 5
Category 017
Appendix A
Coordinate Transformation
Algorithms for the Hand-Over of
Targets Between POEMS
Interrogators
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Document Title
Coordinate Transformation Algorithms for the
Hand-Over of Targets Between POEMS Interrogators
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Cat. 017, Appendix A: Coordinate Transformation
DOCUMENT APPROVAL
The following table identifies all management authorities who have successively approved
the present issue of this document.
AUTHORITY
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DATE
ASTERIX
Manager
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SUR Domain
Manager
J. Berends
SURT
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M. Rees
EATM/DAS
Director
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Cat. 017, Appendix A: Coordinate Transformation
TABLE OF CONTENTS
DOCUMENT IDENTIFICATION SHEET ................................................................................. ii
DOCUMENT APPROVAL ...................................................................................................... iii
DOCUMENT CHANGE RECORD .......................................................................................... iv
TABLE OF CONTENTS .......................................................................................................... v
EXECUTIVE SUMMARY ......................................................................................................... 1
1.
INTRODUCTION
8
1.1
General
8
1.2
Contents
8
1.3
Geometric Constants of the WGS 84 Ellipsoid
8
2.
RADAR LOCAL CO-ORDINATE SYSTEM
9
2.1
Introduction
9
2.2
Radar measurements
9
2.3
Local polar co-ordinates
9
2.4
Conversion from Local Polar Spherical Co-ordinates to Local Cartesian Co-ordinates (and viceversa) 12
3.
GEOCENTRIC CO-ORDINATE SYSTEM
3.1
Introduction
13
13
3.2
Conversion from Local Cartesian Co-ordinates to Geocentric Cartesian Co-ordinates (and viceversa) 13
4.
GEODESIC CO-ORDINATE SYSTEM
4.1
Transformation from Geocentric to Geodesic Co-ordinates
16
4.2
Transformation Geodesic to Geocentric Co-ordinates
18
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5.
SPEED VECTOR
18
6.
SUMMARY OF FORMULAS
19
6.1
Formulas needed for the sending radar station
19
6.2
Formulas needed for the receiving station
20
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EXECUTIVE SUMMARY
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1.
INTRODUCTION
1.1
1.1.1
General
This document has been extracted from a set of documents produced by Smith
System Engineering Limited (Smith) for EUROCONTROL, under contract
C/1.120/HQ/MM/96.
1.2
1.2.1
1.3
1.3.1
Contents
The remainder of this document is structured as follows:
-
section 2 specifies the local co-ordinate system used to measure the aircraft
position in the POEMS system;
-
section 3 specifies and
transformation method;
-
section 4 specifies the transformation to / from geodesic co-ordinates;
-
section 5 specifies the transformation of the speed vector.
the
alternative
absolute
co-ordinate
Geometric Constants of the WGS 84 Ellipsoid
The radar antenna positions and aircraft positions exchanged between mode-S
stations shall be specified in terms of geodetic longitude and latitude, and altitude
relative to the WGS 84 ellipsoid using the following constants.
Name
Page 8
analyses
Notation
Value
Semi-major axis
a
6378137.000 m
Semi-minor axis
b
6356752.314 m
First eccentricity
e
0.0818191908426
(First eccentricity)2
e2
0.00669437999013
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2.
RADAR LOCAL CO-ORDINATE SYSTEM
2.1
Introduction
2.1.1
The measurements made by one radar system are in terms of quantities local to that
radar. This section specifies how these shall be used to obtain the aircraft positions in
terms of local co-ordinates.
2.1.2
Section 2.2 describes the measurements made by the radar system. The specification
of the transformation algorithm required is between local polar co-ordinate systems,
and hence the conversion to these is considered first in section 2.3. In the
consideration of transformation algorithms an intermediate conversion to local
cartesian co-ordinates has been found to be useful, and this is discussed in section
2.3.
2.1.3
The overall problem is conversion between two such local co-ordinate systems
referred with respect to origins at different radar systems. The general approach to be
adopted is transformation to some absolute reference co-ordinate system that is
independent of the radar position, and then from this absolute system to the new local
system.
2.2
Radar measurements
2.2.1
The POEMS system allows the collection of surveillance data that includes both the
position and velocity of aircraft.
2.2.2
The position measurements made by the radar system for a given aircraft are ρ, the
slant range (line-of-sight distance) to the aircraft, θ, the azimuthal bearing of the aircraft
(where approximately due north of the radar is a bearing of 0°), and H, the altitude of
the aircraft above the ground. Note that the first two are measured by the radar itself,
while the third quantity is returned by a transponder on the aircraft. The measurement
of H is made using a pressure altimeter.
2.3
Local polar co-ordinates
2.3.1
The description of the aircraft position in terms of (ρ, θ, H) is a mathematically
unconventional co-ordinate system, which cannot be easily manipulated. It is therefore
helpful to convert to a more convenient system, that of spherical polar co-ordinates
measured relative to the radar system.
2.3.2
The spherical co-ordinate origin is taken to be the radar location, the polar axis is the
normal vertically upwards from the Earth, and the azimuthal zero is a tangent line on
the Earth pointing in the same direction as the zero for measurement of the bearing
angle, as described above. This system is referred to as the “local polar co-ordinate
system”.
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Figure 2-1
Radar measurements of altitude and elevation angle
Figure 2-2
Radar bearing measurement
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2.3.3
In this co-ordinate system the distance from the sphere centre and the azimuthal angle
are already specified by ρ and θ respectively. It only remains to calculate the polar
elevation angle ψ. It should be noted that for compatibility with the notation and
routines used in reference 1, this angle is measured from the azimuthal plane rather
than the polar axis.
2.3.4
The relationship of ρ, θ, H and ψ is illustrated in Figures 2-1 and 2-2.
2.3.5
The radius of the Earth, R, at a given geodetic latitude L is
R=
a (1 − e 2 )
(1 − e 2 sin 2 L) 3
2-1
where a is the semi-major axis of the Earth ellipsoid and e is its eccentricity.
2.3.6
It is generally assumed that R is the same below the aircraft position as at the radar
antenna position. Making this assumption and applying the cosine rule to the triangle
in Figure 2-1 gives the equation:
2 ρ( R1 + h1 )sin ψ = ( H + R1 ) 2 − ( h1 + R1 ) 2 − ρ 2
2-2
which can be rearranged to give an expression for ψ. (Symbols are as defined in
figures 2-1 and 2-2).
2 R1 ( H − h1 ) + H 2 − h12 − ρ 2
sin ψ =
2 ρ( R1 + h1 )
2.3.7
In practice the Earth is actually ellipsoid rather than spherical in shape, bulging slightly
at the equator. Therefore the assumption above introduces an error which can be
approximated as follows. The radius of the Earth changes by about 1 in 300 from the
pole to the equator, i.e. a change of about 20km over a north-south separation of
about 5000km. A radar system is assumed to detect aircraft over a range of about
500km, and hence over this range the Earth’s radius may change by up to 2km.
Therefore the error in the calculated elevation angle is:
∆ψ ≈ tan −1
2.3.8
2-3
2
= 0.2 o
500
2-4
This can be considered to be a systematic feature of representing aircraft positions in
the local polar co-ordinate system - it assumes an Earth of constant radius, equal to
that at the position of the radar.
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2.4
Conversion from Local Polar Spherical Co-ordinates to Local Cartesian
Co-ordinates (and vice-versa)
2.4.1
Standard mathematical formulae can be used to convert from spherical polar coordinates to cartesian co-ordinates relative to the same position.
2.4.2
In accordance with usual convention, the polar axis is taken as z-axis. However, the
due north direction is taken as the y-axis (rather than the conventional choice of the xaxis).
2.4.3
It is understood that the azimuthal measurements from a given radar may not be
measured exactly with respect to the North pole. The deviation of the zero azimuth
direction from North can be denoted as Θ, and used as a correction when making the
conversion to local cartesian co-ordinates, which have been defined with respect to the
exact North direction.
2.4.3
This gives the transformation equations as
x = ρ cos ψ sin(θ + Θ )
y = ρ cos ψ cos(θ + Θ )
z = ρ sin ψ
2.4.4
2-5
This transformation can be inverted using the following equations:
ρ = x2 + y2 + z2
 x
 y
θ = tan −1   − Θ
2-6
 z
ψ = sin −1  
 ρ
2.4.5
This (x,y,z) co-ordinate system is referred to as the “local cartesian co-ordinates”
2.4.6
The conversions between local polar and local cartesian co-ordinates are exact.
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3.
GEOCENTRIC CO-ORDINATE SYSTEM
3.1
Introduction
3.1.1
In this method the absolute reference co-ordinate system is a cartesian grid whose
origin is the centre of the earth. This is referred to as a “geocentric cartesian coordinate system.”
3.1.2
The conversion between local polar and local cartesian co-ordinates was described in
section 2 above.
3.1.3
The conversion between local cartesian and geocentric cartesian co-ordinates is
described in section 3.2 below.
3.2
Conversion from Local Cartesian Co-ordinates to Geocentric Cartesian
Co-ordinates (and vice-versa)
3.2.1
The reference co-ordinate set used is the geocentric co-ordinate system, where the
co-ordinate origin is the centre of the earth, the z-axis points towards the North pole
the x-axis points towards the zero of longitude in the equatorial plane and the y-axis
points towards longitude of 90°E in the equatorial plane. This is illustrated in Figure 31.
Figure 3-1
3.2.2
Geocentric cartesian axes
The local cartesian co-ordinate set has the z-axis as the normal to the Earth’s surface,
the y-axis as a tangent pointing due north, and the x-axis forming the correct right
handed set. These axes are illustrated in Figure 3-2.
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Figure 3-2
Local cartesian axes
3.2.3
The direction vectors of the local cartesian direction axes for radar system 1, X1, Y1,
and Z1, can be written in the geocentric cartesian system.
3.2.4
The position of radar system 1 is specified in terms of a latitude, L1, longitude, G1, and
altitude, h1 (geodetic co-ordinates).
3.2.5
The normal to the Earth’s surface at a given latitude and longitude is then
ZˆS = (cos L1 cos G1 , cos L1 sinG1 , sinL1 )
3-1
and a tangent pointing towards the north is
YˆS = (− sin L1 cosG1 , − sin L1 sin G1 , cos L1 )
3-2
Application of the vector cross product gives:
Xˆ S = (− sinG1 , cosG1 , 0)
3-3
(The notation used is that X$ is a unit vector pointing in the direction in which the coordinate X is being measured)
3.2.6
The position of the radar system is also the co-ordinate origin for the local co-ordinate
system, and is given by:
 (η + h ) cos L1 cos G1 
 1 1

T1 =  ( η1 + h1 ) cos L1 sinG1 


2
 (η 1 (1− e ) + h1 )sin L1 
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where
a
η1 =
3-5
1 − e 2 sin 2 L1
a being the semi-major axis of the Earth ellipsoid and e its eccentricity.
3.2.7
The conversion from the local cartesian system to geocentric cartesian system is
therefore a translation of the origin by T1, and a rotation of the axes from those in
equations 3-1 to 3-3 to the geocentric axes
Xˆ G = (1, 0, 0)
Yˆ = (0, 1, 0)
G
3.2.8
ZˆG = (0, 0, 1)
3-6
X 
X 
 G
 1
T
 YG  = S1  Y 1  +T1
 
 
 ZG 
 Z1 
3-7
Therefore:
where
 − sinG1

S1 =  − sin L1 cosG1

 cos L1 cos G1
3.2.9
cos G1
− sin L1 sin G1
cos L1 sinG1
0 

cos L1 

sin L1 
Note that the matrix S1 is orthogonal, and hence:
S1S1T = I
3.2.10
3-8
3-9
Manipulating equation 3-7 gives the reverse transformation:
X 

X 
 G 

 1
 Y1  = S1   Y G  − T1 
 

 
  ZG 

 Z1 
3-10
3.2.11
It is important to note that all the transformations discussed in this section are in
principle exact.
3.2.12
Therefore the only errors which can arise are from errors in the data put into the
transformation, i.e. the data on radar positions and the aircraft position.
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4.
GEODESIC CO-ORDINATE SYSTEM
4.1
Transformation from Geocentric to Geodesic Co-ordinates
4.1.1
The altitude of the aircraft is already known, from the on-board measurement of H.
However, the latitude and longitude must be calculated from the geocentric
coordinates.
4.1.2
The longitude calculation is straightforward, as this is simply the azimuthal angle in
the geocentric X-Y plane:
tan G =
4.1.3
4-1
The latitude calculation is most easily calculated via the polar angle φ:
tan φ =
4.1.4
YG
.
XG
ZG
X G 2 + YG 2
.
4-2
It is important to note that geodetic latitude, L, is measured as the angle made by a
normal to the surface with the equatorial plane. For an ellipsoid this will differ from
the angle, φ, made with a line joining the point to the centre of the Earth.
φ
L
Figure 4-1 Polar angle (φ) and geodetic latitude (L)
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For a point on the Earth’s surface the two angles are related by:
tan φ = (1 − e 2 ) tan L
or
tan L =
1
tan φ
(1 − e 2 )
4-3
For a point above the earth surface the following formula is valid;
tan L =
1+ H / η
tan φ
(1 − e 2 ) + H / η
4-4
where e is the eccentricity of the reference Earth ellipsoid,
b2
e = 1− 2 ,
a
2
4-5
where b is the semi-minor axis length and a the semi-major axis length of the
ellipsoid.
4.1.5
Using equation 4-3, given that the calculated quantity is the latitude of the point on
the ground directly below the aircraft, the aircraft’s latitude can then be calculated as:
tan L =
4.1.6
1+ H / η
(1 − e 2 ) + H / η
ZG
X G 2 + YG 2
.
4-6
The value of the normalised earth radius at the aircraft latitude η can only be
determined accurately once the latitude is known (see formula 4-7). The next table
investigates the maximum error in case of an approximated normalised earth radius.
normalised earth radius 6378 km
6400 km
Altitude (H)
15 km
η for latitude 0 and 90
1+ H / η
(1 − e 2 ) + H / η
15 km
Difference =
1,00672357744
1,00672363204
-0,0000000546
η shall be approximated by using the normalised earth radius at the radar position,
the resulting error will be smaller than 6.10-8. Note that the error will be less than 1
meter.
Therefore in equation 4-6 the normalised earth radius at the aircraft latitude can be
approximated by :
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η=
a
1 − e 2 sin 2 L1
where L1 is the latitude of the radar.
4.2
Transformation Geodesic to Geocentric Co-ordinates
4.2.1
In this system the aircraft position is specified in terms of geodetic latitude, L,
longitude, G, and altitude, H.
4.2.2
Using normal projection, ie continuing lines of constant latitude normal to the Earth’s
surface, these can be converted to geocentric cartesian coordinates using the
equations below:
XG
YG
ZG
= (η + H ) cos L cos G
= (η + H ) cos L sin G
= (η (1 − e 2 ) + H ) sin L
4-7
where:
η=
5.
a
1 − e 2 sin 2 L
4-8
SPEED VECTOR
5.1.
The speed vector is calculated by the tracking filters and are expressed as Vx and Vy
based on the local Cartesian co-ordinate system used by the tracker. The error in Vx
and Vy strongly depends on sophistication of the applied filtering.
5.2.
The 2D speed in polar co-ordinates as defined in the ASTERIX format can be used in
combination with positions based on local Cartesian or geodesic co-ordinates.
5.3.
If the local tracking is performed in the co-ordinate system specified in section 2, than
the polar speed vector (H = Heading with respect to geographical north at the aircraft
position, Hr = Heading with respect to geographical north of the radar and S =
Ground Speed) shall be calculated as follows.
tan Hr = Vx / Vy
5-1
H = Cf x Hr
5-2
S = V x2 + V y2
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5.4
The correction factor Cf is needed to correct the heading for the difference in
geographical north between radar and aircraft position. Cf depends on the latitude of
the aircraft and difference in longitude between radar and aircraft. Cf could be stored
in a table.
6.
SUMMARY OF FORMULAS
6.1
Formulas needed for the sending radar station
Transform the local polar co-ordinates (ρ, θ, H) from the aircraft to local spherical coordinates (ρ, θ, ψ). R1 and h1 represents the earth radius at the radar position and the
height of the radar.
sin ψ =
2 R1 ( H − h1 ) + H 2 − h12 − ρ 2
2 ρ( R1 + h1 )
2-3
where the earth radius R1 is calculated once using formula 2-1.
Transform the local spherical co-ordinates (ρ, θ, ψ) to local Cartesian co-ordinates
(Xl, Yl, Zl). Θ represents the error in the alignment to the geographical north of the
radar.
X l = ρ cosψ sin(θ + Θ)
Yl = ρ cosψ cos(θ + Θ)
2-5
Z l = ρ sin ψ
Transform the local Cartesian co-ordinates (Xl, Yl, Zl) to geocentric Cartesian coordinates (XG, YG, ZG);
 XG 
 X1

T
 YG = S1  Y 1 + T1
Z
Z
 G
 1
3-7
where the matrices depend on the radar position and are calculated once using
formulas 3-4, 3-8 and 3-9.
Transform the geocentric Cartesian co-ordinates (XG, YG, ZG) to lattitude and
longitude (L, G), the altitude of the aircraft (H) is already known from the beginning;
tan G =
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.
XG
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tan L =
1+ H / η
(1 − e 2 ) + H / η
ZG
X G 2 + YG 2
.
4-6
where η is calculated once with formula below using the lattitude of the radar :
η=
6.2
a
1 − e 2 sin 2 L1
Formulas needed for the receiving station
Transform to latitude and longitude (L, G) and the altitude of the aircraft (H) to the
geocentric Cartesian co-ordinates (XG, YG, ZG);
XG
= (η + H ) cos L cos G
YG
= (η + H ) cos L sin G
ZG
= (η (1 − e ) + H ) sin L
4-7
2
where η depends on the latitude of the aircraft:
η=
a
1 − e 2 sin 2 L
4-8
Transform the geocentric Cartesian co-ordinates (XG, YG, ZG) to local Cartesian coordinates (Xl, Yl, Zl));
X 

X 
 G 

 1
 Y1  = S1   Y G  − T1 
 

 
  ZG 

 Z1 
3-10
where the matrices depend on the radar position and are calculated once using
formulas 3-4, 3-8 and 3-9.
Transform the local Cartesian co-ordinates (x, y, z) to local polar co-ordinates (ρ, θ,
Η). H is already available and Θ represents the error in the alignment to the
geographical north of the radar.
ρ = x2 + y2 + z2
 x
 y
θ = tan −1   − Θ
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